Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:

import java.util.Arrays;
import java.util.Scanner;

public class Main {

    void run() {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        double[] pArr = new double[n];
        for (int i = 0; i < n; i++) {
            pArr[i] = sc.nextDouble();
        }

        // dp[i][j] ... コインをiマイ投げて、jマイ表が出る確率
        double[][] dp = new double[n + 1][n + 1];
        dp[0][0] = 1;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                dp[i + 1][j + 1] = dp[i + 1][j + 1] + dp[i][j] * pArr[i];
                dp[i + 1][j] = dp[i + 1][j] + dp[i][j] * (1 - pArr[i]);
            }
        }
//        debug(dp);
        double ans = 0;
        for (int i = n; i > n / 2; i--) {
            ans += dp[n][i];
        }
        System.out.println(ans);
    }


    void debug(Object... os) {
        System.err.println(Arrays.deepToString(os));
    }

    public static void main(String[] args) {
        new Main().run();
    }
}